# Cold-Plasma Dispersion Relation

It is convenient to define a vector

 (5.40)

that points in the same direction as the wavevector, , and whose magnitude, , is the refractive index (i.e., the ratio of the velocity of light in vacuum to the phase-velocity). Equation (5.9) can be rewritten

 (5.41)

Without loss of generality, we can assume that the equilibrium magnetic field is directed along the -axis, and that the wavevector, , lies in the - plane. Let be the angle subtended between and . The eigenmode equation (5.41) can be written

 (5.42)

The condition for a nontrivial solution is that the determinant of the square matrix be zero. With the help of the identity

 (5.43)

we find that (Hazeltine and Waelbroeck 2004)

 (5.44)

where

 (5.45) (5.46) (5.47)

The dispersion relation (5.44) is evidently a quadratic in , with two roots. The solution can be written

 (5.48)

where

 (5.49)

Note that . It follows that is always real, which implies that is either purely real, or purely imaginary. In other words, the cold-plasma dispersion relation describes waves that either propagate without evanescense, or decay without spatial oscillation. The two roots of opposite sign for , corresponding to a particular root for , simply describe waves of the same type propagating, or decaying, in opposite directions.

The dispersion relation (5.44) can also be written

 (5.50)

For the special case of wave propagation parallel to the magnetic field (i.e., ), the previous expression reduces to

 (5.51) (5.52) (5.53)

Likewise, for the special case of propagation perpendicular to the field (i.e., ), Equation (5.50) yields

 (5.54) (5.55)